Prove the converse part of the proof of Theorem 4.17. That is, let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of \(d\)-dimensional random vectors and let \(\mathbf{X}\) be a \(d\)-dimensional random vector. Prove that if \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{X}\) as \(n ightarrow \infty\) then \(\mathbf{v}^{\prime} \mathbf{X}_{n} \xrightarrow{d} \mathbf{v}^{\prime} \mathbf{X}\) as \(n ightarrow \infty\) for all \(\mathbf{v} \in \mathbb{R}^{d}\).

Transcribed Image Text:

Theorem 4.17 (Cramr and Wold). Let {X}1 be a sequence of random vectors in Rd and let X be a d-dimensional random vector. Then XX as n if and only if v'X v'X as n for all v Rd.